San José State University

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Thayer Watkins
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The Potential, Kinetic and Total Energies of the Planets

Let M be the mass of the Sun and m the mass of a planet and R its distance from the Sun. The potential energy of the planet is then

V = −GMm/R

The kinetic energy K of a planet is ½mv², where v is the planet's tangential velocity. The total energy E which is of interest is K+V. A planet also has rotational kinetic energy that is not included. The energy E is the amount of energy that would be required to remove a planet from our solar system.

The notation (nE+m) stands for n×10m.

The Energies of the Planets
(joules)
Planet Potential
Energy
Kinetic
Energy
Total
Energy
Mercury -7.5526E+32 3.79059E+32 -3.76201E+32
Venus -5.98571E+33 2.9749E+33 -3.01081E+33
Earth -5.29201E+33 2.66762E+33 -2.62439E+33
Mars -3.73775E+32 1.86979E+32 -1.86796E+32
Jupiter -3.24179E+35 1.62046E+35 -1.62133E+35
Saturn -5.28556E+34 2.65372E+34 -2.63184E+34
Uranus -4.01788E+33 2.00961E+33 -2.00827E+33
Neptune -3.02974E+33 1.5216E+33 -1.50814E+33

The current global energy use by the human race is approximately 5.5×1020 joules per year. The energy required to remove Earth from the Solar System is then equal to about 5 trillion years of human energy use. The amount of energy required to remove Jupiter is about 24 times that required for the Earth, but given Jupiter's great size that is surprisingly little.

One notes that the magnitude of the kinetic energy of a planet is about one half of the magnitude of its potential energy. This can be derived.

There is a balance between gravitational force and centrifugal force for a planet; i.e.,

GMm/R² = mv²/R

The balance is independent of the mass of the planet. Thus

GM/R² = v²/R
and hence
v² = GM/R

Therefore

K = ½mv² = ½GMm/R = ½|V|

The ratios of K to |V| should have been precise one half, but because of approximations in the data used they were not.

An additional bit of analysis using the above definitions gives Kepler's Law, the relationship between the orbit period T and the orbit radius R.

The period T is (2πR/v) thus

T = (2πR)/[GM/R]½ = [2π/(GM)½]R3/2
or, equivalently
T² = (4π²/(GM))R³

This is Kepler's Law.


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